We have introduced a novel approach to competency-based education in mechatronics from the undergraduate to the postgraduate level. What distinguishes this approach is the integration of modeling and control of sampled systems right from the beginning of the undergraduate education. It is achieved by changing the structure of the first-semester Calculus course to focus on discrete-time systems: emphasizing numerical differentiation and integration and difference equations. The curriculum is enriched by interdisciplinary homework project assignments that, given in subsequent courses throughout the education, are tied to the same quarter-vehicle model but vary in theoretical complexity. It demonstrates multiple dimensions of the same engineering problem, leading to a deeper understanding. Based on the discrete-time modeling studied in Calculus, students can solve the problem at a basic level and verify the results with measurements. Later, they can compare these solutions with those obtained using more advanced tools. This approach creates a synergy between different subjects ranging from the basics to the advanced control theory. This article focuses primarily on the mathematical toolkit that facilitates the achievement of our didactic goals.